System and Method for Characterizing Ferromagnetic Material

ABSTRACT

A system and method using magnetic sensing to non-intrusively and non-destructively characterize ferromagnetic material within infrastructure. The system includes sensors for measuring magnetic field gradients from a standoff distance adjacent to ferromagnetic material. The method includes using the system to measure magnetic fields, determining magnetic field gradients measured by a sensor array, and comparing measured and modeled or historical magnetic field gradients at the same or similar positions to identify differences caused by a phenomenon in the ferromagnetic material, and, in a particular embodiment, to recognize defects and developing defects.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of U.S. patent application Ser. No.15/653,036, filed Jul. 18, 2017, entitled “System and Method forCharacterizing ferromagnetic Material,” which is a continuation of U.S.patent application Ser. No. 15/197,699, filed Jun. 29, 2016, entitled“System and Method for Characterizing ferromagnetic Material,” whichclaims the benefit of U.S. Provisional Patent Application No.62/185,888, filed Jun. 29, 2015, entitled “Detection of Defects inFerromagnetic Materials using Large Standoff Magnetization (LSM)Sensors,” and U.S. Provisional Patent Application No. 62/265,851, filedDec. 10, 2015, entitled “System and Method for CharacterizingFerromagnetic Material,” each of which is hereby incorporated byreference herewith, in its entirety, for all purposes.

BACKGROUND ART

Metal components of structures are susceptible to defects, such as dueto imperfect manufacture, corrosion, fatigue, wear, damage, etc. Toprevent catastrophic failure of such structures, metal components may bevisually inspected to identify defects before a failure occurs. However,many structures are not easily inspected due to being buried undergroundor beneath the sea, or due to being embedded within other materials suchas concrete. For large infrastructure that contains metal components,visual inspection may be impractical or too costly to perform routinely.

Many ferromagnetic objects, including steel pipe, act as weak permanentmagnets even when not intentionally magnetized; for example, magneticdipoles in steel may partially orient to the Earth's magnetic fieldafter cooling below the Curie temperature when cast or hot-rolled in thefoundry. Magnetic fields present in ferromagnetic objects as straybyproducts of their manufacture are known herein as parasitic fields.The Earth's magnetic field also induces magnetic fields in ferromagneticobjects. These magnetic fields permit detection of ferromagnetic objectsfrom a distance. Magnetic exploders for naval mines and torpedoes havebeen designed to detect magnetic fields from large ferrous objects, suchas warships, since 1917, although both German and American magneticexploders were problematic when used in combat on torpedoes in1939-1943. Magnetic exploders, however, are merely intended to detectthe object from a distance, not to detect or analyze defects in thatobject.

Magnetic particle inspection is well known as a method for detectingcracks in objects. In this technique, a ferromagnetic object is placedin a magnetic field, and magnetic particles, such as iron filings, areapplied to the object. The magnetic field may be provided by passing anelectric current through the object, or by placing the object in a fieldprovided by an electromagnet. If a crack is present, the magneticparticles cluster near the crack. Field strengths used for magneticparticle inspection are typically much greater than the Earth's magneticfield, or those parasitic fields that may be present in ferromagneticmaterials.

SUMMARY OF EMBODIMENTS

According to an embodiment, a method for characterizing a ferromagneticmaterial includes: receiving measured magnetic field data from aplurality of sensors adjacent the ferromagnetic material at a pluralityof locations along the ferromagnetic material; deriving measuredmagnetic field features from the measured magnetic field data; comparingthe derived magnetic field features with modeled or previouslycollected, verified magnetic field features to identify differencescaused by a phenomenon in the ferromagnetic material.

According to another embodiment, a system for characterizing aferromagnetic material includes: memory capable of storing magneticfield data from at least one sensor configured to measure magnetic fielddata at a plurality of scan positions along the ferromagnetic material,and software including machine readable instructions. The system mayfurther include a processor coupled with the memory, the processorconfigured to, in response to execution of the software, perform thesteps of: derive magnetic field feature data from the magnetic fielddata at the plurality of scan positions, and compare the measuredmagnetic field features data with modeled magnetic field feature data toidentify a phenomenon in the ferromagnetic material.

BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWING

The invention will be more fully understood by referring to thefollowing Detailed Description of Specific Embodiments in conjunctionwith the Drawings, of which:

FIG. 1 is a block diagram of one system for characterizing ferromagneticmaterial, in an embodiment.

FIG. 2 is a block diagram of another system for characterizingferromagnetic material, in an embodiment.

FIG. 3 illustrates one magnetic sensor array used in a system forcharacterizing ferromagnetic material, in an embodiment.

FIG. 4 illustrates a system for characterizing ferromagnetic material,in an embodiment.

FIG. 5 illustrates a pipe made of ferromagnetic material.

FIG. 6 is a flowchart including steps of a method to characterizeferromagnetic material, in an embodiment.

FIG. 7 shows a plot of magnetic field versus scan position from onesensor in a system that characterizes ferromagnetic material.

FIG. 8 illustrates a plot of measured magnetic field strength versusscan position for a system that characterizes ferromagnetic material.

FIG. 9 illustrates a plot of dipole model magnetic field strength versusscan position for a system that characterizes ferromagnetic material.

FIG. 9A and FIG. 9B represent plots of magnetic field gradients versusscan position in presence of a weld.

FIG. 9C and FIG. 9D represent plots of magnetic field gradients versusscan position in presence of a defect.

FIG. 10 shows a plot of magnetic field strength versus scan position foran axial dipole model, in an embodiment.

FIG. 11 shows a plot of magnetic field strength versus scan position fora lateral dipole model, in an embodiment.

FIG. 12 shows a plot of magnetic field strength versus scan position fora vertical dipole model, in an embodiment.

FIG. 13 shows a plot of magnetic field strength versus scan position fora combination dipole model, in an embodiment.

FIG. 14 is a flowchart illustrating steps of a method to characterize aferromagnetic material, in an embodiment.

FIG. 15 shows a plot of measured magnetic field gradients versus scanposition, in an embodiment.

FIG. 16 shows a plot of dipole model magnetic field gradients versusscan position, in an embodiment.

FIG. 17 shows steps for determining magnetic field gradients in FIG. 14,in an embodiment.

FIG. 18 is a flow chart for a method to identify a phenomenon withinferromagnetic material, in an embodiment.

FIG. 19 shows a pairwise statistical comparison plot for characterizingferromagnetic material, in an embodiment.

FIG. 20A, FIG. 20B, and FIG. 20C show diagrams of schemes for combiningmagnetic field data with data from other sensing modalities, in anembodiment.

DETAILED DESCRIPTION OF SPECIFIC EMBODIMENTS

FIG. 1 schematically illustrates one system 100 for characterizing aferromagnetic material 130, in embodiments. System 100 non-intrusivelyand non-destructively detects local phenomena in an infrastructure,including defects and non-defects, based on ferromagnetic material 130.System 100 includes a plurality of magnetic sensors 101. Although FIG. 1shows four magnetic sensors 101, system 100 may have more or fewersensors 101 without departing from the scope hereof. Sensors 101 coupleto a data processing module 150 via communication paths 115, which mayinclude one or both of a wired and/or a wireless communication media.Data processing module 150 processes magnetic field measurementsreceived from sensors 101 via communication paths 115 to characterizeferromagnetic material 130 as described below. Data processing module150 has at least one processor 152 coupled with a memory 154, and may insome embodiments have a global positioning system (GPS) receiver 156and/or a digital-radio uplink 158. Digital-radio uplink 158 may operatethrough a cell phone network, or other wireless network such as WiFi,for example, to transmit or receive information to a server 160. Server160 may include, in embodiments, a database 162 of anomalies.

Ferromagnetic material 130 exhibits magnetization based on itsstructure, composition, and fabrication history. At the same time,ferromagnetic material 130 may have a phenomenon 135 that perturbs themagnetic field from ferromagnetic material 130, as illustrated bymagnetic field lines 140 in FIG. 1, wherein phenomenon 135 “disrupts” anotherwise spatially regular magnetic field of ferromagnetic material130. Phenomenon 135 is for example (a) a weld or junction betweensegments of ferromagnetic material 130, (b) an unintentionalirregularity of cracked, missing or otherwise faulty ferromagneticmaterial (hereinafter called a “defect” and typically due to corrosion,fatigue, wear, damage or imperfect manufacture; some defects may lead toinfrastructure failure), or (c) an intentionally-designed gap oropening. Sensors 101 may be magnetometers arranged in an array tomeasure magnetic field 140 related to phenomenon 135.

Identifying a defect in material 130 prior to failure in components suchas reinforcing steel, pipelines, oil platform legs, ship hulls, etceteraburied underground or located underwater often requires inspectingbeneath a visible surface. The embodiments disclosed herein may besuitable in evaluating ferromagnetic material of such infrastructureincluding, but not limited to: industrial vessels and pipes of plantsand equipment, including power plants, refineries and heat exchangers;pipelines, such as oil and gas pipelines; railways, including rails andbridges of railroads, light-rail and subways; structures, such asbuildings and bridges made with ferrous beams or rebar-reinforcedconcrete; and partially or fully submerged drilling rigs, ships andsubmarines.

During use of system 100 to inspect infrastructure, system 100 ispositioned near, and moved along ferromagnetic material 130 while system100 measures material-associated magnetic field 140. Sensors 101 arearranged in a spatially distributed array that provides a spatial map ofmagnetic field 140, at each traveled location along material 130, witheach sensor 101 measuring both magnetic direction and magnitude. Dataprocessing module 150 in turn processes magnetic field measurementsreceived from the array of sensors 101 via communication paths 115 tocharacterize magnetic field 140, thereby providing a current scan ofmagnetic field along ferromagnetic material 130.

In data processing system 150, processor 152 may execute software (forexample software 263 discussed in further detail below with respect toFIG. 2), realized as machine readable instructions stored in memory 154,to implement (a) scan routines to store the current scan of the magneticfield in memory 152, and (b) analysis routines to analyze the scan ofthe magnetic field for anomalies such as phenomenon 135. If an anomalyis located, processor 152 may further execute additional software (forexample software 263 discussed in further detail below), also realizedas machine readable instructions stored in memory 154, to implementfurther analysis routines on the stored scan to identify the anomaly asa non-defect, such as a weld, flange, or intentionally-designedgap/opening, or identify the anomaly as a defect, such as a missingmetal defect or other unintentional fault within the material 130. Itshould be appreciated that various aspects of data processing system 150may be performed remotely, such as in server 160, without departing fromthe scope hereof. For example, the analysis routines, includinganalyzing the scan of magnetic field for anomalies such as phenomenon135, may be performed on a scan that is previously implemented by dataprocessing module 150 (via scan routines) and then transmitted from dataprocessing module 150 to server 160.

In embodiments, the analysis routines operate by determining signaturephenomena, of the observed magnetic field (such as phenomena in, orfunctions of, the magnetic field gradients and derivatives thereof) asrecorded from multiple locations in a sliding window of the scan. In anembodiment, the software (for example software 263 discussed in furtherdetail below) implementing such analysis routines determines signaturephenomena by fitting a superposition of predefined signature phenomena.The predefined signature phenomena may be derived from (a) computermodels of magnetic dipoles to the observed magnetic field from thelocations in the sliding window, (b) a non-dipole based model, (c)measurements, or (d) a combination thereof.

Information about anomaly types, including classifications of theanomaly types and pattern phenomena corresponding to each anomaly type,may be stored in memory 154 and/or database 162. In an embodimentoptimized for analysis of pipelines, the anomaly types include exemplarygood welds and exemplary defective welds, as well as cracks, breaks,valves, taps, and corroded locations. The analysis routines may beconfigured to provide the classification that most closely matches eachanomaly found during a scan.

A location read from GPS 156, and/or other location sensors such as anodometer, may in some embodiments be associated with a portion of thescan associated with a defect, or in some embodiments portions of thescan associated with a non-defect, such as a weld or flange, and theselocations and associated scan windows are reported through uplink 158 toserver 160 and stored in database 162. Since weld locations in apipeline, or bolted joints in railroad track, are unlikely to changewith time in infrastructure 130, new phenomena, or phenomena that havesignificantly changed character since any prior scan, can indicateincipient failure such as cracks in a pipe or breaks in rail. Eitherprocessor 152 or server 160, may correlate the current and prior scan toalign phenomena, and then compare phenomena of anomalies detected in thecurrent scan to observations made during a prior scan at the samelocation, as may have been previously recorded in database 162, todetermine whether the phenomenon is new, and identify it as new. Newphenomena, as well as phenomena classified as defects, may warrantfurther investigation, such as by excavating a pipeline.

In particular embodiments, system 100 does not include a bias magnet formagnetizing the ferromagnetic material 130. In these embodiments, themagnetic fields sensed by system 100 are parasitic magnetic fields andfields induced in the ferromagnetic material by the Earth's magneticfield.

FIG. 2 schematically illustrates a system 200 that characterizesferromagnetic material. System 200 is a an embodiment of system 100 Insystem 200, sensors 101, of FIG. 1, are implemented in a sensor array250 that communicatively couples to data processing module 150. System200 implements data processing module 150 with at least one processor264 in communication with memory 262. Processor 264 is an embodiment ofprocessor 152. Memory 262 is an embodiment of memory 154 and may betransitory and/or non-transitory and in some embodiments includes one orboth of (a) volatile memory such as RAM and (b) non-volatile memory suchas, ROM, EEPROM, Flash-EEPROM, magnetic media including disk drives,optical media. Memory 262 stores software 263 and firmware 261 asmachine readable instructions executable by processor 264 to processdata from sensor array 250 and identify and/or characterize one or morephenomena 135 of ferromagnetic material 130. It should be appreciatedthat various aspects of software 263 and firmware 261 may be implementedby server 160 shown in FIG. 1, instead of, or in addition to, dataprocessing module 150. In embodiments, measurements from sensor array250 are received by a receiver 255 that communicates measurements todata processing module 150. In other embodiments, measurements arecommunicated directly from sensor array 250 to data processing module150. Receiver 255 is for example a data acquisition device. Inembodiments, data from non-magnetic sensors 252 (e.g., accelerometers)are also received by receiver 255, as more fully described below.Illustratively, data processing module 150 includes an interface 265 forcommunicating with other devices, including server 270 that processesand stores data. Server 270 is similar to server 160, and therefore thediscussion of server 160 applies equally to server 270. Although dataprocessing module 150 is shown as a single device, it should beappreciated that data processing module 150 may incorporate one or moredevices such as computers, processors, memories, etc.

FIG. 3 schematically illustrates an exemplary magnetic sensor array 300for characterizing ferromagnetic material 330 in the form of a pipe.Sensor array 300 includes ten magnetic sensors, including a firstmagnetic sensor 301, a magnetic second sensor 302, and so on up to atenth magnetic sensor 310 arranged in a three-dimensional (3D) array.More or fewer magnetic sensors may be utilized without departing fromthe scope hereof. Sensor array 300 is an embodiment of sensor array 250,FIG. 2, and each sensor 301-310 is for example an embodiment of sensor101 of FIGS. 1-2. FIG. 3 illustrates an exemplary “T” arrangement ofsensors 301-310 positioned along three orthogonally oriented axes.

Although in FIG. 3, sensor array 300 is shown in a “T” arrangement, thesensor array 300 may be configured in other patterns without departingfrom the scope hereof. For example, sensor array 300 may also beimplemented in non-orthogonal arrangements, instead of the orthogonalarrangement shown in FIG. 3 without departing from the scope hereof.Moreover, sensor array 300, in either a non-orthogonal or an orthogonalarrangement may be configured with more or fewer magnetic sensors, andcould be deployed in positions and arranged in a pattern, such as in acone- or sphere-shaped pattern. Furthermore, in embodiments, the sensorarray may be synthesized with just a single magnetic sensor movedbetween known positions to make multiple measurements as a data array.Likewise, the locations of sensors 301-310 need not be restricted tolocations along the axes of a 3D coordinate system. One- ortwo-dimensional arrays may also be beneficially employed as array 300.

Magnetic sensor array 300 is positioned with a standoff distance 312above ferromagnetic material 330 having a defect 350. Ferromagneticmaterial 330 is an example of ferromagnetic material 130, FIG. 1, whiledefect 350 is an example of phenomenon 135. Defect 350 is for example amissing metal defect, a corrosion-induced defect, or any other type ofirregularity that is substantially different from an expected shape andstructure of ferromagnetic material 330. Defect 350 thus causes amagnetic field phenomenon with an exemplary magnetization directionindicated by arrow 340. Standoff distance 312 may be known, estimated ormeasured, for example using ground penetrating radar.

The ability to sense magnetic fields with sensor arrays, such as sensorarray 300, depends on standoff distance 312, the strength of magneticfield 340 from ferromagnetic material 330, the sensitivity of magneticsensors 301-310, and spacing distances 321, 322, 323, 324, 325 betweensensors 301-310 in sensor array 300. In an embodiment, magnetic sensors301-310 are magnetometers that measure magnetic fields. Magnetic sensors301-310 may be one-axis magnetometers that measure magnetic fields alongone axis, two-axis magnetometers that measure magnetic fields along twoaxes, or three-axis magnetometers that measure magnetic fields alongthree axes. The three axes are for example x, y, and z axes depicted inFIG. 3. Note that sensor array 300 includes variable spacing distancesbetween magnetic sensors 301-310; for example, a first distance 321between magnetic sensors 301 and 302 is greater than a second distance322 between magnetic sensors 302 and 303. Similarly along the z-axis, afourth distance 324 may be greater than a fifth distance 325. In anembodiment, first, second, third, fourth, and fifth distances 321, 322,323, 324, 325 are optimized to measure dipole magnetic fields anddetermine magnetic field gradient peak signatures of defect 350 for agiven standoff distance 312. In an operational example, which theembodiments herein are not limited to, a third distance 323 betweenmagnetic sensors 303 and 304 is about 15 cm for a standoff distance 312of 25 cm. In another operational example, magnetic sensors 301-310 haveadjustable positions within sensor array 300 such that sensor spacingdistances 321, 322, 323, 324, 325 are adjusted to optimize measurementof magnetic fields having different field strengths for differentstandoff distances 312.

FIG. 4 illustrates yet another system 400 for characterizingferromagnetic material 430. System 400 is an embodiment of system 100.System 400 shows four sensor arms 411, 412, 413, 414 each of whichcontains one or more sensors 410 (e.g., magnetometers) that measuremagnetic field strength. Sensors 410 may be arranged in an array, suchas the sensor array 300 of FIG. 3, and attached to a frame 420 by sensorarms 411, 412, 413, 414 or by other structure when moving the array ofsensors 410 along ferromagnetic material 430. Sensors 410 are anembodiment of sensors 101 and are arranged in an example of sensor array250. Ferromagnetic material 430 is an example of ferromagnetic material130. By way of example, frame 420 may be equipped with straps 405 orother means for a user to carry system 400. In another embodiment,system 400 is mechanically coupled to a vehicle, such as an automobile,train, aerial vehicle, or underwater vehicle. Sensor arms 411, 412, 413,414 may be moveable up and down along frame 420 to account for variationin standoff distance 312.

A power supply 440 electrically couples to sensors 410 to provide directcurrent (DC) electrical power. Power supply 440 may be wired to anelectrical grid or have a battery pack that enables remote, off-grid useof system 400. A receiver 455 couples to sensors 410 via communicationpath 415, which is similar to communication path 115 of FIG. 1, toreceive data therefrom. Receiver 455 is for example an embodiment ofreceiver 255, FIG. 2. A computer 460 connects to receiver 455 viacommunication path 425 to process received sensor data. Computer 460 isfor example an embodiment of data processing module 150 implementingprocessor(s) 264, memory 262, and optional interface 265. Communicationpaths 415, 425 may include one or both of a wired and/or a wirelesscommunication media.

FIG. 5 shows an exemplary pipe 530 made of ferromagnetic material. Pipe530 is an example of ferromagnetic material 130 and 430 and may becharacterized using any of systems 100, 200, and 400. Pipe 530 includesa weld 535, which is a welded junction that joins a first segment 531 toa second segment 532 of pipe 530. Weld 535 is an example of anintentional non-defect phenomenon that produces a characteristicmagnetic field phenomenon providing a magnetic field signature that mayresemble a magnetic dipole. For example, magnetic flux leakage may occurat weld 535 producing the magnetic field signature. In an embodiment,magnetic field signatures are determined in real-time and used forcalibration and compensation of magnetic field measurements caused byvariability such as platform motion or standoff distance 312. Dataprocessing module 150 compares magnetic field measurements obtained bysensors 101 to known magnetic field signatures to detect a defect, suchas defect 450, FIG. 4. In FIG. 2, memory 262 may thus include at leastone magnetic field signature for this purpose.

FIG. 6 is a flowchart illustrating steps of an exemplary method 600 formeasuring magnetic field 140 from infrastructure containingferromagnetic material 130. Method 600 is an example of a “scan routine”as discussed above with respect to FIGS. 1, 2, and 4. As such, method600 may be performed by system 100 of FIG. 1, system 200 of FIG. 2, andsystem 400 of FIG. 4, for example using data processing module 150executing software 263.

In an optional step 610, the system for characterizing ferromagneticmaterial moves to a first scan position, such as an arbitrary locationadjacent to infrastructure containing ferromagnetic material. In anexample of step 610, system 400 of FIG. 4 is moved to a positionadjacent to first segment 531 of pipe 530 of FIG. 5. In other examples,system 100 or 200, of FIGS. 1 and 2, is moved to a position adjacent toa first segment of ferromagnetic material 130.

In a step 620, the system measures magnetic fields. In an example ofstep 620, sensors 410 measure a magnetic field (e.g. magnetic field 140)from first segment 531. In other examples of step 620, sensors 110 ofFIGS. 1-2, possibly in the arrangement of array 300 of FIG. 3, measure amagnetic field from ferromagnetic material 130.

In a step 630, the system for characterizing ferromagnetic materialmoves to a next scan position. In an example of step 630, system 400 ofFIG. 4 moves to a position adjacent weld 535 of pipe 530 of FIG. 5. Inanother example of step 630, system 100 or 200, of FIG. 1-2, is moved toa next scan position along ferromagnetic material 130.

Step 640 is a decision. If in step 640 the end of the infrastructure isreached, or the end of a desired scan range is reached, method 600 ends.Otherwise, method 600 returns to step 620. In this way, method 600 iscarried out to scan an entire infrastructure or a desired portion of aninfrastructure. The rate at which magnetic fields are measured betweenfirst scan position and the next scan position may depend on bandwidthof data acquisition such as receiver 455 of FIG. 4. In an embodiment,system 400 is moved between locations at a rate of 0.25 meters persecond. In another embodiment, system 100 or 200, of FIG. 1-2, is movedat a rate of 0.25 meters per second along ferromagnetic material 130.

FIG. 7 shows a plot 700 of exemplary magnetic fields measured by onesensor, such as sensor 304 in array 300 or any of sensors 110 of FIGS.1-2 and 410 of FIG. 4, versus scan position along pipe 330.Specifically, plot 700 illustrates exemplary magnetic field 140 measuredby this sensor during method 600 over multiple iterations of step 620.Plot 700 includes magnitude of magnetic field, B, aligned in x, y, and zaxes (B_(x), B_(y), B_(z)) versus scan position along pipe 330. Adataset 710 shows magnetic field strength along the x-axis, B_(x),versus scan position; a dataset 720 shows magnetic field strength alongthe y-axis, B_(y), versus scan position; and a dataset 730 showsmagnetic field strength along the z-axis, B_(z), versus scan position.The scan direction is oriented along the x-axis and sensor 304 iscentered above pipe 330 in the y-dimension. By way of comparison, at ascan position of zero in FIG. 7, sensor 304 of FIG. 3 is positioneddirectly above defect 350. As sensor 304 is moved along ferromagneticmaterial 330, the scan position from defect 350 varies, corresponding toan increasing (positive values) or decreasing (negative values) scanposition depending on the direction of movement.

Referring again to FIG. 6, in an optional step 650, magnetic field datameasured from step 620 is processed to characterize ferromagneticmaterial 130. In an example of step 650, measured magnetic field data iscompared, by data processing module 150 executing software 263 (oralternatively a remote server such as server 160 executing softwaresimilar to software 263), with an empirically determined orphysics-based model of magnetic fields to identify and characterizephenomena in the magnetic field data caused by a phenomenon offerromagnetic material 130. Measured data and modeled data are comparedusing for example matched filters or statistical-detection algorithms.One example of a physics-based model is a magnetic dipole model. Missingmetal from ferromagnetic material produces predominantly magnetic dipolecharacteristics that are detected and matched with a magnetic dipolemodel. Missing metal defects, such as defect 350, FIG. 3, may have adipole in reverse orientation to magnetization in ferromagnetic material330. The reverse dipole orientation may be used to help identify defect350. Similarly, welds forming junctions between segments offerromagnetic material, such as weld 535 of FIG. 5, producepredominantly magnetic dipole characteristics. For example, at weld 535between pipe segments 531, 532 dipoles may exist due to differences inmagnetization direction and amplitude between pipe segments 531, 532together with magnetic reorientation due to heating when the weld wasmade.

In a particular embodiment, modeled data is determined from a finiteelement model. In embodiments, model-based analysis, for exampleperformed by data processing module 150 executing software 263, ofmagnetic dipoles detected by the system includes one or more of:applying interpolation on the magnetic field signature sphere to obtainthe magnetic field at planes above and parallel and near-parallel to thepipe at different distances, and angles; extracting magnetic fieldspatial phenomena from the magnetic field, such as gradient, directionalderivative, divergence or Laplacian, curl, magnitude and neighborhoodlocal statistical moments of these phenomenon fields; obtaining daughtermagnetic field phenomena from the field, such as a Spatial Fast-FourierTransform (FFT) phase field, power spectral density (PSD), and Waveletcoefficients; separately analyzing each phenomenon statistically, forexample using the t-test and the Wilcoxon Rank test; and selectingphenomena by collectively satisfying, or optimally satisfying, multiplecriteria such as p-values, correlation to size and height, andorthogonality (non-correlation among phenomena). Nearby pairs andtriplets of the above phenomena are fused for FFT and Wavelet analysis.Extracted phenomena are compared to a library of model-derivedphenomena, such as welds and defects.

FIGS. 8 and 9 show exemplary plots of measured and modeled magneticfield strength, respectively, as a function of scan position. FIG. 8shows a plot 800 of exemplary magnetic fields measured by a singlesensor (e.g. one of sensors of array 300) for a range of scan positionsusing method 600 of FIG. 6 implemented by system 400 of FIG. 4. Plot 800may thus illustrate magnetic fields at a plurality of scan positions forweld 535 of FIG. 5 such as measured with sensor 304 for example. Adataset 810 shows magnetic field strength along the x-axis, B_(x), adataset 820 shows magnetic field strength along the y-axis, B_(y), and adataset 830 shows magnetic field strength along the z-axis, B_(z), overa range of scan positions along the x-axis at a position centered overthe pipe in the y-dimension. Magnetic field strength of pipe segments asdetermined at weld 535, such as that illustrated in plot 800, may beused for scaling magnetic field measurements from pipe segments 531, 532to normalize data for improved detection of defects.

FIG. 9 shows a plot 900 of exemplary magnetic field strength versus scanposition from a dipole model used in characterizing a ferromagneticmaterial phenomenon, such as weld 535 that joins first and second pipesegments 531, 532 of FIG. 5. A dataset 910 shows magnetic field strengthalong the x-axis, B_(x), a dataset 920 shows magnetic field strengthalong the y-axis, B_(y), and a dataset 930 shows magnetic field strengthalong the z-axis, B_(z), versus scan position along the x-axis at aposition centered over the pipe in the y-dimension.

According to an embodiment, data processing module 150 compares measuredmagnetic field plots, such as plot 800 of FIG. 8, with modeled magneticfield plots, such as plot 900 of FIG. 9 to distinguish a weld signaturefrom a defect signature in step 650 of method 600, thereby detectingwhether a defect has occurred. While both weld and defect signatureshave dipole characteristics, magnetic field changes along the pipe maydiffer in magnitude from those expected at a weld. Further, fieldgradients at a weld will tend to taper from a field orientation in onesegment of the pipe to a potentially-different orientation in anothersegment of the pipe, rather than returning to the same orientationbeyond the defect as to be expected in a single section of pipe. Thismay be due to a broad transition zone between magnetic polarization ofpipe sections produced as the metal was heated and cooled duringwelding, this transition zone being broader than typical missing metaldefects.

In an embodiment, a scalar likelihood, L, indicates the presence of adefect determined from gradients in all axes in a scan position windownear the phenomenon, and from other statistical processing; if L isgreater than a threshold, the phenomenon or anomaly is reported as adefect. FIGS. 9A-9D illustrate L plotted versus scan window positionwith a threshold of one; FIGS. 9A and 9B are associated with weldsignatures and L<1 indicating non-defect, for FIGS. 9C and 9D, L>1indicating a defect. The window size may be varied and the gradient datarescanned repeatedly with different window sizes depending on the sizesof ferromagnetic material, phenomenon (e.g. defects, weld, or anomaly)as discussed further below with respect to FIG. 18.

Magnetic fields calculated from dipole models for x, y and z-axes, suchas those plotted versus scan position in FIG. 9, depend on orientationof the magnetic dipole. For example, a dipole may have an axialorientation along the scanning direction, for example along the x-axisof FIG. 3, a lateral orientation sideways from the scanning orientation,for example along the y-axis of FIG. 3, or a vertical orientation thatis up and down from the scanning direction, for example along the z-axisof FIG. 3. A combination dipole has magnetization components of allthree orientations, C_(x), C_(y), and C_(z). Three-axis magnetic fieldsare calculated for a dipole model using Equation 1, below.

$\begin{matrix}{\begin{pmatrix}B_{x} \\B_{y} \\B_{z}\end{pmatrix} = {\frac{1}{r^{5}}\begin{pmatrix}{{C_{x}\left( {{3\; x^{2}} - r^{2}} \right)} + {3\; C_{y}{xy}} + {3\; C_{z}{xz}}} \\{{3\; C_{x}{xy}} + {C_{y}\left( {{3\; y^{2}} - r^{2}} \right)} + {3\; C_{z}{yz}}} \\{{3\; C_{x}{xz}} + {3\; C_{y}{yz}} + {C_{z}\left( {{3\; z^{2}} - r^{2}} \right)}}\end{pmatrix}}} & (1)\end{matrix}$

Equation 1 is the magnetic field equation for an arbitrary dipoleorientation where C_(x) C_(y), and C_(z) are combination magnetic fieldsproportional to magnetization along the x, y, and z-axes, respectively,and r is the absolute distance that includes standoff distance 312 fromthe sensor to the magnetic field source. In order for a magneticsignature to resemble a dipole, sensor distance from a magnetic source,r, is for example about two to three times longer than the magneticsource itself, although shorter sensor distances contain dipolecharacteristics that may be matched to Equation 1 if r is known.

FIG. 10 shows a plot 1000 of exemplary modeled magnetic fields versusscan position for an axial dipole model, aligned with the x-axis, whichmay be used by data processing module 150 (or server 160 implementinganalysis functions) to identify phenomena of ferromagnetic material 130.A dataset 1010 shows magnetic field strength along the x-axis, B_(x), adataset 1020 shows magnetic field strength along the y-axis, B_(y), anda dataset 1030 shows magnetic field strength along the z-axis, B_(z),for a magnetic dipole source oriented axially. G is a constant, C_(y)and C_(z) are zero. Each of datasets 1010, 1020, and 1030 show themagnetic field as a function of x at a position centered over themagnetic dipole source in the y-direction.

FIG. 11 shows a plot 1100 of exemplary modeled magnetic field strengthversus scan position for a lateral dipole model, aligned with they-axis, which may be used by data processing module 150 (or server 160implementing analysis functions) to identify phenomena of ferromagneticmaterial 130. A dataset 1110 shows magnetic field strength along thex-axis, B_(x), a dataset 1120 shows magnetic field strength along they-axis, B_(y), and a dataset 1130 shows magnetic field strength alongthe z-axis, B_(z) for magnetic dipole source oriented laterally. C_(y)is a constant, C_(x) and C_(z) are zero. Each of datasets 1110, 1120,and 1130 show the magnetic field as a function of x at a positioncentered over the magnetic dipole source in the y-direction.

FIG. 12 shows a plot 1200 of exemplary modeled magnetic field strengthversus scan position for a vertical dipole model which may be used bydata processing module to identify phenomena of ferromagnetic material130. A dataset 1210 shows magnetic field strength along the x-axis,B_(x), a dataset 1220 shows magnetic field strength along the y-axis,B_(y), and a dataset 1230 shows magnetic field strength along thez-axis, B_(z) for a magnetic dipole source oriented vertically. C_(z) isa constant, C_(x) and C_(y) are zero. Each of datasets 1210, 1220, and1230 show the magnetic field as a function of x at a position centeredover the magnetic dipole source in the y-direction.

FIG. 13 shows a plot 1300 of exemplary combination magnetic fieldstrength versus scan position, which combines axial, lateral, andvertical dipole orientations of FIGS. 10-12. A dataset 1310 showsmagnetic field strength along the x-axis, B_(x), a dataset 1320 showsmagnetic field strength along the y-axis, B_(y), and a dataset 1330shows magnetic field strength along the z-axis, B_(z). C_(x), C_(y) andC_(z) are constants adjusted for model fitting, based on factorsincluding the strength of measured magnetic fields.

Other than comparing models and measurements of magnetic fields overscan position, such as step 650 of method 600, magnetic field gradientsmay be used to further identify phenomena of ferromagnetic material 130.According to an embodiment, magnetic field gradients are calculated froma plurality of sensors arranged in an array, such as sensor array 300 ofFIG. 3. Specifically, FIG. 14 is a flowchart illustrating steps of onemethod 1400 to detect a phenomenon of a ferromagnetic material andcharacterize the ferromagnetic material based upon magnetic field dataobtained using one or more sensors. Each sensor (e.g. sensors 110, 310,410) is configured to measure the magnitude and direction of the localmagnetic field. Method 1400 uses models and measurements of magneticfields over scan position to detect and characterize phenomenon 135 offerromagnetic material 130. Data processing module 150 (or server 160implementing analysis functions) may perform method 1400 based uponmagnetic field data obtained from sensor array 250. Method 1400 may beimplemented in data processing module 150 (or server 160) as at least aportion of software 263 and/or firmware 261, FIG. 2. Accordingly, itshould be appreciated that method 1400 may also be implemented usingsystem 400, of FIG. 4. Aspects of method 1400 are for example anembodiment of step 650 of method 600.

In step 1410, magnetic field data are received for a plurality of scanpositions. In an example of step 1410, processor 264 executes software263 and/or firmware 261 stored in memory 262 to parse data from sensorarray 250, which is received either directly from sensor array 250 oroptionally via receiver 255.

In step 1420, magnetic field derived features are derived from themagnetic field data of step 1410. Exemplary magnetic field derivedfeatures comprise numerics that are derived from the raw sensor data, ora denoised version thereof, including but not limited to: the fieldmeasurements, their Fourier, Wavelet or any other transform, theirmagnetic field gradients; the gradient Fourier transform, wavelettransform or any other transform; 2^(nd) derivative matrices orHessians, their Fourier transforms or any of their transforms, fractaldimension of the field, gradients, Hessians, or features recovered bydata mining or machine learning/deep learning methods.

In an example of step 1420, the magnetic field derived features that arecalculated are magnetic field gradients. In such example, the magneticfield gradients are calculated, by data processing module 150 (or server160), from differences in magnetic fields between sensors 301-310 ofsensor array 300, FIG. 3 for a plurality of scan positions. In oneembodiment, a single sensor such as sensor 304 measures magnetic fieldsat a plurality of scan positions, and one or more gradients arecalculated, using for example data processing module 150 (or server160), from the plurality of measurements. In another embodiment,magnetic field gradients between different sensors are calculated foreach scan position. Equation 2, below, shows an exemplary calculationfor magnetic field gradients between fourth sensor 304 and eighth sensor308 along the x-axis of FIG. 3.

$\begin{matrix}{\frac{\Delta \; B_{xyz}}{\Delta \; x} = {\begin{pmatrix}{B_{x_{S\; 4}} - B_{x_{S\; 8}}} \\{B_{y_{S\; 4}} - B_{y_{S\; 8}}} \\{B_{z_{S\; 4}} - B_{z_{S\; 8}}}\end{pmatrix}/x_{{S\; 4} - {S\; 8}}}} & (2)\end{matrix}$

In Equation 2, ΔB_(xyz)/Δx is the difference between three-axis magneticfields between sensor 304 (abbreviated S4) at position x_(S4) and sensor308 (abbreviated S8) at position x_(S8). B_(xS4) is the x-axis magneticfield at fourth sensor 304, B_(xS8) is the x-axis magnetic field ateighth sensor 308, and so on for y-axis and z-axis magnetic fields,B_(y), B_(z). x_(S4-S8) is the spacing distance between sensors 304 and308.

Three-axis magnetic field gradients are calculated from dipole models ofmagnetic fields for additional select pairs of sensors in the samemanner. For example, three-axis magnetic field gradients (ΔB_(xyz)) arecalculated using Equation 3, below, between fourth sensor 304 and ninthsensor 309, between fourth sensor 304 and tenth sensor 310, and betweenninth sensor 309 and tenth sensor 310 along the z-axis, as depicted inFIG. 3.

$\begin{matrix}{\frac{\Delta \; B_{xyz}}{\Delta \; z} = \begin{pmatrix}\frac{B_{x_{S\; 4}} - B_{x_{S\; 9}}}{z_{{S\; 4} - {S\; 9}}} & \frac{B_{x_{S\; 4}} - B_{x_{S\; 10}}}{z_{{S\; 4} - {S\; 10}}} & \frac{B_{x_{S\; 9}} - B_{x_{S\; 10}}}{z_{{S\; 9} - {S\; 10}}} \\\frac{B_{y_{S\; 4}} - B_{y_{S\; 9}}}{z_{{S\; 4} - {S\; 9}}} & \frac{B_{y_{S\; 4}} - B_{y_{S\; 10}}}{z_{{S\; 4} - {S\; 10}}} & \frac{B_{y_{S\; 9}} - B_{y_{S\; 10}}}{z_{{S\; 9} - {S\; 10}}} \\\frac{B_{z_{S\; 4}} - B_{z_{S\; 9}}}{z_{{S\; 4} - {S\; 9}}} & \frac{B_{z_{S\; 4}} - B_{z_{S\; 10}}}{z_{{S\; 4} - {S\; 10}}} & \frac{B_{z_{S\; 9}} - B_{z_{S\; 10}}}{z_{{S\; 4} - {S\; 10}}}\end{pmatrix}} & (3)\end{matrix}$

In Equation 3, ΔB_(xyz)/Δz is the difference between three-axis magneticfields along the z-axis, z_(S4-S9) is the spacing distance betweenfourth sensor 304 (abbreviated S4) and ninth sensor 309 (abbreviatedS9), B_(xS4) is the x-axis magnetic field at fourth sensor 304, B_(xS9)is the x-axis magnetic field at ninth sensor 309, and so on for othersensor pairs and for y-axis and z-axis magnetic fields, B_(y), B_(z).

Similarly, select three-axis magnetic field gradients (ΔB_(xyz)/Δy) arecalculated along the y-axis using Equation 4, below.

$\begin{matrix}{\frac{\Delta \; B_{xyz}}{\Delta \; y} = \begin{pmatrix}\frac{B_{x_{S\; 1}} - B_{x_{S\; 2}}}{y_{S\; 1} - y_{S\; 2}} & \frac{B_{y_{S\; 1}} - B_{y_{S\; 2}}}{y_{S\; 1} - y_{S\; 2}} & \frac{B_{z_{S\; 1}} - B_{z_{S\; 2}}}{y_{S\; 1} - y_{S\; 2}} \\\frac{B_{x_{S\; 1}} - B_{x_{S\; 3}}}{y_{S\; 1} - y_{S\; 3}} & \frac{B_{y_{S\; 1}} - B_{y_{S\; 3}}}{y_{S\; 1} - y_{S\; 3}} & \frac{B_{z_{S\; 1}} - B_{z_{S\; 3}}}{y_{S\; 1} - y_{S\; 3}} \\\frac{B_{x_{S\; 1}} - B_{x_{S\; 4}}}{y_{S\; 1} - y_{S\; 4}} & \frac{B_{y_{S\; 1}} - B_{y_{S\; 4}}}{y_{S\; 1} - y_{S\; 4}} & \frac{B_{z_{S\; 1}} - B_{z_{S\; 4}}}{y_{S\; 1} - y_{S\; 4}} \\\frac{B_{x_{S\; 2}} - B_{x_{S\; 3}}}{y_{S\; 2} - y_{S\; 3}} & \frac{B_{y_{S\; 2}} - B_{y_{S\; 3}}}{y_{S\; 2} - y_{S\; 3}} & \frac{B_{z_{S\; 2}} - B_{z_{S\; 3}}}{y_{S\; 2} - y_{S\; 3}} \\\frac{B_{x_{S\; 1}} - B_{x_{S\; 5}}}{y_{S\; 1} - y_{S\; 5}} & \frac{B_{y_{S\; 1}} - B_{y_{S\; 5}}}{y_{S\; 1} - y_{S\; 5}} & \frac{B_{z_{S\; 1}} - B_{z_{S\; 5}}}{y_{S\; 1} - y_{S\; 5}} \\\frac{B_{x_{S\; 2}} - B_{x_{S\; 4}}}{y_{S\; 2} - y_{S\; 4}} & \frac{B_{y_{S\; 2}} - B_{y_{S\; 4}}}{y_{S\; 2} - y_{S\; 4}} & \frac{B_{z_{S\; 2}} - B_{z_{S\; 4}}}{y_{S\; 2} - y_{S\; 4}} \\\frac{B_{x_{S\; 1}} - B_{x_{S\; 6}}}{y_{S\; 1} - y_{S\; 6}} & \frac{B_{y_{S\; 1}} - B_{y_{S\; 6}}}{y_{S\; 1} - y_{S\; 6}} & \frac{B_{z_{S\; 1}} - B_{z_{S\; 6}}}{y_{S\; 1} - y_{S\; 6}} \\\frac{B_{x_{S\; 2}} - B_{x_{S\; 5}}}{y_{S\; 2} - y_{S\; 5}} & \frac{B_{y_{S\; 2}} - B_{y_{S\; 5}}}{y_{S\; 2} - y_{S\; 5}} & \frac{B_{z_{S\; 2}} - B_{z_{S\; 5}}}{y_{S\; 2} - y_{S\; 5}} \\\frac{B_{x_{S\; 3}} - B_{x_{S\; 4}}}{y_{S\; 3} - y_{S\; 4}} & \frac{B_{y_{S\; 3}} - B_{y_{S\; 4}}}{y_{S\; 3} - y_{S\; 4}} & \frac{B_{z_{S\; 3}} - B_{z_{S\; 4}}}{y_{S\; 3} - y_{S\; 4}} \\\frac{B_{x_{S\; 1}} - B_{x_{S\; 7}}}{y_{S\; 1} - y_{S\; 7}} & \frac{B_{y_{S\; 1}} - B_{y_{S\; 7}}}{y_{S\; 1} - y_{S\; 7}} & \frac{B_{z_{S\; 1}} - B_{z_{S\; 7}}}{y_{S\; 1} - y_{S\; 7}} \\\frac{B_{x_{S\; 2}} - B_{x_{S\; 6}}}{y_{S\; 2} - y_{S\; 6}} & \frac{B_{y_{S\; 2}} - B_{y_{S\; 6}}}{y_{S\; 2} - y_{S\; 6}} & \frac{B_{z_{S\; 2}} - B_{z_{S\; 6}}}{y_{S\; 2} - y_{S\; 6}} \\\frac{B_{x_{S\; 3}} - B_{x_{S\; 5}}}{y_{S\; 3} - y_{S\; 5}} & \frac{B_{y_{S\; 3}} - B_{y_{S\; 5}}}{y_{S\; 3} - y_{S\; 5}} & \frac{B_{z_{S\; 3}} - B_{z_{S\; 5}}}{y_{S\; 3} - y_{S\; 5}} \\\frac{B_{x_{S\; 3}} - B_{x_{S\; 6}}}{y_{S\; 3} - y_{S\; 6}} & \frac{B_{y_{S\; 3}} - B_{y_{S\; 6}}}{y_{S\; 3} - y_{S\; 6}} & \frac{B_{z_{S\; 3}} - B_{z_{S\; 6}}}{y_{S\; 3} - y_{S\; 6}} \\\frac{B_{x_{S\; 4}} - B_{x_{S\; 5}}}{y_{S\; 4} - y_{S\; 5}} & \frac{B_{y_{S\; 4}} - B_{y_{S\; 5}}}{y_{S\; 4} - y_{S\; 5}} & \frac{B_{z_{S\; 4}} - B_{z_{S\; 5}}}{y_{S\; 4} - y_{S\; 5}} \\\frac{B_{x_{S\; 2}} - B_{x_{S\; 7}}}{y_{S\; 2} - y_{S\; 7}} & \frac{B_{y_{S\; 2}} - B_{y_{S\; 7}}}{y_{S\; 2} - y_{S\; 7}} & \frac{B_{z_{S\; 2}} - B_{z_{S\; 7}}}{y_{S\; 2} - y_{S\; 7}} \\\frac{B_{x_{S\; 4}} - B_{x_{S\; 6}}}{y_{S\; 4} - y_{S\; 6}} & \frac{B_{y_{S\; 4}} - B_{y_{S\; 6}}}{y_{S\; 4} - y_{S\; 6}} & \frac{B_{z_{S\; 4}} - B_{z_{S\; 6}}}{y_{S\; 4} - y_{S\; 6}} \\\frac{B_{x_{S\; 3}} - B_{x_{S\; 7}}}{y_{S\; 3} - y_{S\; 7}} & \frac{B_{y_{S\; 3}} - B_{y_{S\; 7}}}{y_{S\; 3} - y_{S\; 7}} & \frac{B_{z_{S\; 3}} - B_{z_{S\; 7}}}{y_{S\; 3} - y_{S\; 7}} \\\frac{B_{x_{S\; 5}} - B_{x_{S\; 6}}}{y_{S\; 5} - y_{S\; 6}} & \frac{B_{y_{S\; 5}} - B_{y_{S\; 6}}}{y_{S\; 5} - y_{S\; 6}} & \frac{B_{z_{S\; 5}} - B_{z_{S\; 6}}}{y_{S\; 5} - y_{S\; 6}} \\\frac{B_{x_{S\; 4}} - B_{x_{S\; 7}}}{y_{S\; 4} - y_{S\; 7}} & \frac{B_{y_{S\; 4}} - B_{y_{S\; 7}}}{y_{S\; 4} - y_{S\; 7}} & \frac{B_{z_{S\; 4}} - B_{z_{S\; 7}}}{y_{S\; 4} - y_{S\; 7}} \\\frac{B_{x_{S\; 5}} - B_{x_{S\; 7}}}{y_{S\; 5} - y_{S\; 7}} & \frac{B_{y_{S\; 5}} - B_{y_{S\; 7}}}{y_{S\; 5} - y_{S\; 7}} & \frac{B_{z_{S\; 5}} - B_{z_{S\; 7}}}{y_{S\; 5} - y_{S\; 7}} \\\frac{B_{x_{S\; 6}} - B_{x_{S\; 7}}}{y_{S\; 6} - y_{S\; 7}} & \frac{B_{y_{S\; 6}} - B_{y_{S\; 7}}}{y_{S\; 6} - y_{S\; 7}} & \frac{B_{z_{S\; 6}} - B_{z_{S\; 7}}}{y_{S\; 6} - y_{S\; 7}}\end{pmatrix}} & (4)\end{matrix}$

In Equation 4, ΔB_(xyz)/Δy is the difference between three-axis magneticfields along the y-axis, y_(S1-s2) is the spacing distance between firstsensor 301 (abbreviated S1) and second sensor 302 (abbreviated S2),B_(xS1) is the x-axis magnetic field at first sensor 301, B_(xS2) is thex-axis magnetic field at second sensor 302, and so on for other sensorpairs and for y-axis and z-axis magnetic fields, B_(y) and B_(z).

In an example of step 1420, x-axis magnetic field gradients(ΔB_(xyz)/Δx) are calculated using Equation 2 from differences betweenthree-axis magnetic fields (B_(x), B_(y), B_(z)) measured with fourthsensor 304 (S4) and eighth sensor 308 (S8) along the x-axis as depictedin FIG. 3. Similarly, select z-axis magnetic field gradients(ΔB_(xyz)/Δz) are calculated using Equation 3 for magnetic fieldsmeasured with fourth sensor 304 (S4), ninth sensor 309 (S9), and tenthsensor 310 (S10), along the z-axis, as depicted in FIG. 3. Similarly,select y-axis magnetic field gradients (ΔB_(xyz)/Δy) are calculatedusing Equation 4 for magnetic fields measured with first sensor 301(S1), second sensor 302 (S2), third sensor 303 (S3), fourth sensor 304(S4), fifth sensor 305 (S5), sixth sensor 306 (S6), and seventh sensor307 (S7), along the y-axis, as depicted in FIG. 3. Exemplary measuredmagnetic field gradients are plotted in FIG. 15.

FIG. 15 shows a plot 1500 of exemplary measured magnetic field gradientsin the x-axis versus scan position. Plot 1500 is determined by dataprocessing module 150 from magnetic field 140 of defect 350 measuredusing sensor array 300 of FIG. 3 for example. Dataset 1510 shows a firstgradient ΔB_(x) between first sensor 301 and second sensor 302. Dataset1520 shows a second gradient ΔB_(x) between first sensor 301 and thirdsensor 303. Dataset 1530 shows a third gradient ΔB_(x) between firstsensor 301 and fourth sensor 304. Dataset 1540 shows a fourth gradientΔB_(x) between third sensor 303 and fourth sensor 304. Dataset 1550shows a fifth gradient ΔB_(x) between third sensor 303 and fifth sensor305.

Although step 1420 is described above including measured magnetic fieldgradients, it should be appreciated that other measured magnetic fieldderived features (other than gradients) could be utilized in step 1420.For example, instead of gradients, step 1420 may calculate measuredmagnetic field hessians, wavelets, power spectral density, or fractaldimension without departing from the scope hereof. As such, it should beappreciated that, although equations 2-4 above show the formula forgradients, step 1420 may be implemented based on similar formulas formany other magnetic field derived features that are derived from themagnetic field sensor data, such as those magnetic field derivedfeatures discussed above.

In an embodiment, method 1400 includes optional step 1430, wherein atleast one model of magnetic field derived features is calculated frommodeled magnetic fields for a plurality of scan positions. In an exampleof step 1430, modeled magnetic field gradients shown in FIG. 16 arecalculated by data processing module 150 (or server 160) using Equations2-4 from model magnetic fields calculated using Equation 1 for selectpairs of sensors and a plurality of scan positions. In an alternativeembodiment, modeled magnetic field features are calculated fromhistorical data as found in database 162 for the same location. Forexample, if the same ferromagnetic material 130 was previously scannedusing method 600, the measured magnetic field features are used as amodel for comparison with repeat measurements. This approach enables (a)monitoring a small anomaly that may be a defect over time to determineif it is growing in size; growth in size is more likely associated witha developing defect than with a weld or flange.

FIG. 16 shows a plot 1600 of exemplary magnetic field gradients in thex-axis versus scan position calculated by data processing module 150 (orserver 160) for a dipole model of a defect, such as defect 350 of FIG.3. Dataset 1610 shows a first gradient ΔB_(x) between first sensor 301and second sensor 302. Dataset 1620 shows a second gradient ΔB_(x)between first sensor 301 and third sensor 303. Dataset 1630 shows athird gradient ΔB_(x) between first sensor 301 and fourth sensor 304.Dataset 1640 shows a fourth gradient ΔB_(x) between third sensor 303 andfourth sensor 304. Dataset 1650 shows a fifth gradient ΔB_(x) betweenthird sensor 303 and fifth sensor 305. Again, it should be appreciatedthat step 1430 is not limited to magnetic field gradients, but can beimplemented based on other magnetic field derived features such as thosediscussed above.

In step 1440, measured magnetic field derived feature data are comparedto modeled magnetic field feature data for a plurality of scan positionsto identify one or more phenomena in magnetic field features caused bywelds, defects, or anomalies in the ferromagnetic material. In anexample of step 1440, multiple measured magnetic field gradients fromsensor array 300, such as those shown in FIG. 15, are compared, usingdata processing module 150 (or server 160), to modeled magnetic fieldgradients, such as those shown in FIG. 16, to identify a phenomenon inmagnetic field gradients caused by defect 350 of ferromagnetic material330 of FIG. 3. As part of step 1440, measured and modeled data may beanalyzed for correct dipole orientation based on dipole model gradients.

According to an embodiment, select magnetic field phenomena containing adefect signature are used to identify defect 350. According to anotherembodiment, step 1440 includes an optional step 1442 of incorporatingdata from non-magnetic sensors 252 of FIG. 2 to further enhancecharacterization of ferromagnetic material 130. In one example,non-magnetic sensors 252 provide ground penetrating radar used tomeasure standoff distance 312. In another example, data processingmodule 150 utilizes GPS location information provided by GPS 156 foreach magnetic field measurement, which may be augmented by one or bothof Wide Area Augmentation System (WAAS) data and odometer data.

In an optional step 1450, one or more defects or irregularities of aferromagnetic material are characterized, and their locations andclassifications may be reported in step 1460. In an example of step1450, defect 350 of FIG. 3 is identified and characterized. In anexample of step 1460, location of defect 350 is reported to server 160and stored in database 162, FIG. 1. Reporting location of defects andirregularities includes displaying two and three-dimensional plots oninterface 265 of data processing module for example. Depending on thetype of phenomenon identified, a more intrusive inspection, such asdigging out an underground pipe for visual inspection, may be performedin the identified locations.

Characterization of a defect by data processing module 150 in step 1450may include determining its size and orientation, and may furtherinclude classifying a type of missing metal defect. Characterization mayinclude distinguishing between a defect and a non-defect such as a weld,flange, coupled branch line, bend, or other normal or intentionalanomaly. Identification and characterization of defects andirregularities may be assisted using information from different sensortypes and prior magnetic sensor data for the same location. Method 1400provides advantages for identifying and characterizing phenomena inferromagnetic material including that the method may be automated and isrepeatable.

FIG. 17 shows an exemplary method 1700 for determining a model, and thusa signature, for observed magnetic field gradients. Method 1700 is anembodiment of aspects of FIG. 14.

In one embodiment, method 1700 includes a step 1710 of plotting magneticfield data for a plurality of locations and a plurality of sensors viainterface 265 for analysis by a user to determine a nearest sensor to amagnetic field source. For example, plot 800 of FIG. 8 may be analyzedfor a weld signature from measurements made of weld 535 of FIG. 5 usingmethod 600 of FIG. 6. Step 1710 may occur in method 1400 prior to step1410.

In a step 1720, a nearest sensor of the sensor array to a phenomenon ofthe ferromagnetic material is determined. In one embodiment, dataprocessing module 150 determines the nearest sensor. In an example ofthis embodiment of step 1720, processor 264 executes a portion ofsoftware 263 and/or firmware 261 to process magnetic field datagenerated by sensors 301-310 of FIG. 3 to determine that sensor 304 ofFIG. 3 is nearest defect 350. In another embodiment, a user identifiessensor 304 as the nearest sensor to defect 350 by visually inspectingmagnetic field plots displayed in step 1710. Step 1720 may occur inmethod 1400 between steps 1410 and 1420.

In a step 1730, magnetic field data from the nearest sensor, measuredover a plurality of scan positions, are analyzed for known signatures.In an example of step 1730, using data processing module 150, magneticfield data from nearest sensor 304 of FIG. 3 are analyzed for signaturesof one or more known phenomena in ferromagnetic material, such as weld535 of FIG. 5. In an embodiment of step 1730, measured magnetic fieldsversus scan position along the ferromagnetic material, such as in plot800 of FIG. 8, are compared to a magnetic dipole model versus scanposition, such as in plot 900 of FIG. 9. In an embodiment of step 1730,known signatures are analyzed via data processing module 150 usingmatched filters and statistical-detection algorithms.

If a signature is found in step 1730, a step 1740 isolated a portion ofthe magnetic field data that matches a known signature. In an example ofstep 1740, using data processing module 150, magnetic field datacorresponding to a weld signature from weld 535 of FIG. 5 are isolatedfrom magnetic field data of first and second pipe segments 531, 532.According to an embodiment, a user crops magnetic field data using dataprocessing module 150 to isolate a weld signature. For example, plot 800of FIG. 8 may be cropped between scan positions to a narrower windowranging from −1.7 m to 1.8 m to isolate the weld signature.

Steps 1730 and 1740 may occur in method 1400 between steps 1440 and1450. For example, if steps 1730 and 1740 are used in method 1400, step1730 may act to filter out known non-defects (such as welds) from thephenomenon identified in step 1440. Steps 1730 and 1740 may utilizenon-magnetic sensors, such as GPS, and ground penetrating radar, asdiscussed above with respect to step 1442 to further enhanceidentification of known non-defects in method 1700.

In a step 1750, a characterization is determined for the segment offerromagnetic material having a phenomenon. Step 1750 acts to identifythe phenomenon as defects, and then potentially characterize saididentified phenomenon as a specific type of defect. The characterizationand phenomenon location are then reported in step 1460, FIG. 14. In anexample of step 1750, using data processing module 150, magnetic fluxleakage at weld 535 of pipe 530, FIG. 5 is analyzed to determine amagnetization direction and a magnetization amplitude (or strength) forfirst and second segments 531, 532.

In an embodiment, using data processing module 150 (or server 160),modeled magnetic data is modeled as a linear subspace of components ofthe magnetic signal over scan position, such as gradients, wavelets, andpower spectral density. The magnetic signal components are extractedfrom a physics-based model, such as a dipole model, and corrupted withnoise and interference from one or more magnetic sources to make themodel more realistic. Magnetic sensor measurements are then projectedonto the subspace spanned by dipole moments, or any function of themagnetic dipole moments, such as gradients, Hessians, wavelets, powerspectral density, or fractal dimension of other magnetic field derivedfeatures discussed above. Equation 5a shows an example linear subspacemodel.

X=Sθ+Fφ+U _(ψ) +n  (5a)

In Equation 5a, X is a gradient measurement vector across scanpositions, S is a feature subspace basis matrix across scan positions interms of gradients, F is a known magnetic interference subspace such asa bias or flange, U is an unknown magnetic interference subspace matrix,n is a noise vector, and θ, φ, and ψ are scaling parameter vectorsdetermined from measurements. U may be constructed as the matrixorthogonal to a concatenation of S and F.

Again, it should be appreciated that X may represent feature measurementvectors other than gradient. For example, within Equation 5a, thesubspace basis matrix S is based on gradients, but it should beappreciated that the subspace basis matrix S may be based on othermagnetic field measurements such as those magnetic field derivedfeatures discussed above. In an embodiment, subspace basis matrix S isphysics dipole moment based. In this embodiment, the phenomena ofinterest within the measured magnetic field data are made of dipoles(geometric shapes discussed above), with a varying magnitude (small vs.large defects, defects vs. weld, etc.) In another embodiment, thesubspace basis matrix S is constructed based on learning techniques suchas Singular Value Decomposition (SVD), Espirit, and Music algorithms.

Equation 5a linearly models the phenomenon identified within themagnetic field raw data. Using equation 5a, data processing module 150(or server 160) can both identify and characterize a detected phenomenonwithin the measured magnetic field data. For example, within dataprocessing module 150 (or server 160) and using equation 5a, for a givenphenomenon, a window size W is selected. Within that window, magneticfield derived features are determined. The window size W may be adjustedfor sensitivity to features of different sizes. For example, a smallwindow size W may be used to aim detection at small-scale features,whereas a larger window size W may be used to aim detection atlarger-scale features. In another example, the same dataset may beanalyzes using two or more different window sizes to be sensitive tofeatures of a variety of sizes. In the above example of gradients,computations of equations 2-4, over the determined window W, derivedfrom all possible pairs of sensor measurements, provide the canonicalshape of what a gradient of the magnetic field for any event looks like.Equation 5a's modulation by the vector θ determines whether a dipolemoment based phenomenon is present. If the magnitude of θ is above athreshold, then the phenomena contains a defect (or in other words adefect is detected). The direction of the vector θ may be utilized tocharacterize the phenomena, as discussed below. 4) The matrix Frepresents other known events that may be non-dipole moment based, ordifferent. F is computed as in equation 3.

It should be appreciated that non-linear models may be utilized insteadof the linear model shown in equation 5a. For example, non-linear modelswould include an equation 5b.

X=S(θ)+F(φ)+n  (5b)

S, F are a non-linear function of θ, ϕ. Under equation 5b, either S, F,or both, may be learned using non-linear curve fitting, neural networks,deep-learning algorithms, etc. For each phenomenon within the measuredmagnetic field data, S (or F) may have its own shape.

A hypothesis test may be used to determine whether the measured magneticfield data does not (null hypothesis, H0) or does (first hypothesis, H1)include a phenomenon signature that is a defect. Equations 6 and 7 statean exemplary hypothesis test based on equation 5a, but may be modifiedas understood by those of ordinary skill based on equation 5b, above.

H0: X=Fφ+Nψ+n  (6)

Equation 6 shows null hypothesis, H0, which states that the gradientmeasurement vector across scan positions, X, is due to (a) knowninterference subspace, F, plus (b) a subspace N which is the subspaceorthogonal to the projection of subspace S onto the subspace orthogonalto known interference subspace F, and (c) noise vector n. Herein, eachof F, N, and S interchangeably refers to the respective matrix as wellas the subspace spanned by the columns of the matrix.

H1: X=Sθ+Fφ+n  (7)

Equation 7 shows first hypothesis, H1, which states that the gradientmeasurement vector across scan positions, X, is due to feature subspacebasis matrix across scan positions in terms of gradients, S, plus knowninterference subspace, F, and noise vector n.

The output of the hypothesis test of Equations 6 and 7 is a statisticproportional to the likelihood, L, of a phenomenon being present.Equations 6 and 7 may be graphically understood with respect to FIGS.9A-D, where hypothesis H0 is shown in FIGS. 9A and 9B because only weldsare shown and the likelihood never crosses threshold. By contrast, FIGS.9C-D show hypothesis H1 because defects 1 and 2 are shown and thelikelihood crosses the threshold.

Thus, it is shown that a defect may be identified in a binary manner(e.g. presence versus absence of defect, but not yet classified todetermine the type of defect). The likelihood compares the observedvalue X of equation 5 to a threshold. This decision may be made byselecting the most likely event, which is the phenomenon in a dictionaryof phenomena that most closely resembles the measurement X, preferably(but not necessarily) after accounting for noise in the data. Thisdecision may utilize a hypothesis test, as shown in equations 6 and 7,or alternatively/additionally, a nearest neighbor model, or any otherpattern classification/machine learning/deep-learning algorithm. Tocompensate for noise, statistic used thereby may be a Chi-Squarestatistic, an F statistic, or non-Gaussian generalization of theChi-Square or F statistic such as those discussed in: MN Desai, R SMangoubi, “Robust Gaussian and non-Gaussian matched subspace detection,”IEEE Transactions on Signal Processing, 2003.

It should be appreciated that functions other than the likelihoodfunction may be utilized, such as the robust likelihood function whichis a trimmed version of the likelihood function that protects againstnoise outliers. Moreover, the estimate of θ, ϕ, or {circumflex over(θ)}, {circumflex over (ϕ)}, may be obtained by inverting the matrix orfunctions (non-linear models) S, F, respectively. The magnitude anddirection of these vectors may then be used instead of the likelihoodfunction. Embodiments where the noise model is unknown and thenon-parametric approach is used, may use non-parametric statistics suchas the sign test, the rank sum test, rank histograms of the noise, etc.

The magnitude of phenomenon scaling parameter vector, θ, may be astatistic for determining the presence of a phenomenon, the size of thephenomenon, and the magnetization direction of the phenomenon.

In an embodiment, modeled magnetic data is modeled as a non-linearsubspace of components of the magnetic signal versus scan position, suchas a polynomial, neural network, or learning-based technique, fitted toa measured magnetic field data curve. The coefficients of the non-linearsubspace may include components that determine the presence of phenomenaand characterize the nature of those phenomena. In another embodiment, afractal dimension of the measured magnetic field data is used todetermine the presence of phenomena and to characterize the nature ofthose phenomena.

It should be appreciated that the models of Eq. 5a and 5b may bereplaced by models not based on feature subspaces S and F.

FIG. 18 is a flowchart for a method 1800 to identify a phenomenon withinferromagnetic material by comparing modeled and magnetic field data overa variable window of scan positions. In step 1810, method 1800 comparesmodeled and magnetic field data, such as gradient data, from a smallwindow of scan positions corresponding to a portion of a ferromagneticmaterial. In one example of operation of step 1800, data processingmodule 150 compares modeled magnetic field data to captured magneticfield data, captured using one or more of sensors 350 of FIG. 3,corresponding to a window of scan positions along ferromagnetic material130. Method 1800 is an example of steps 1440-1450 and 1750 of FIGS. 14and 17, respectively.

Step 1820 is a decision. If step 1820 determines that a likelihood, L,has crossed a predefined threshold indicating that a phenomenon ispresent in the ferromagnetic material, then method 1800 proceeds withstep 1860. Otherwise, method 1800 proceeds with step 1830 to increasewindow size. In an example of step 1820, L has crossed a predefinedlikelihood threshold of for example one (L>1), as shown in FIGS. 9C and9D, indicating presence of a defect within a scan position window fromzero to one along the x-axis. In another example of step 1820, L has notcrossed the predefined threshold of one (L<1) in a scan window from zeroto one, as shown in FIGS. 9A and 9B, indicating absence of a defect. Thepredefined likelihood threshold may take on other values than one,without departing from the scope thereof. For example, the predefinedlikelihood threshold may depend on whether or not the likelihood L hasbeen normalized and the nature of such normalization. Step 1820 is anexample of step 1450 and 1750 of methods 1400 and 1700, respectively.

In optional step 1830, the window size is increased. In an example ofstep 1830, the window for comparing measured and modeled magnetic fielddata is increased to the entire range of zero to two shown in FIGS.9A-9D. Window as used herein means the number of data pointssurrounding, or beginning from, a given scan position in the measuredmagnetic field data.

Step 1840 is a decision. If, in step 1840, the window size has beenincreased to maximum, method 1800 proceeds to step 1850, whichdetermines that no defect is present in the corresponding portion offerromagnetic material. Otherwise, method 1800 returns step 1820 todetermine if the likelihood threshold has been crossed. In an example ofstep 1840, the window size corresponds to scan positions taken alongfirst segment 531 of pipe 530, FIG. 5, which is not a maximum window andmethod 1800 returns to step 1820. Steps 1830 to 1860 together form anexample of step 1440 of method 1400.

In step 1860, a magnetic field source is identified. In an example ofstep 1860, a magnetic field phenomenon is identified from defect 450,FIG. 4.

Step 1870 is a decision. If in step 1870, a large window is determinedto have been used, then control passes to step 1880 where a non-defectis determined. In an example of step 1830, a window covering scanpositions for first and second pipe segments 531, 532 of FIG. 5 was usedand the magnetic source identified in step 1860 was from weld 535.Otherwise, if a large window was not used, for example the windowincludes data from only first pipe segment 531, method 1800 proceeds tostep 1890, which determines that a defect is present within the scanpositions of the ferromagnetic material corresponding to the window.Steps 1860 and 1870 are examples of step 1450 of FIG. 14.

Method 1800 uses data windows and may apply steps 1820 to 1840repeatedly to identify phenomena having different sizes. For example,method 1800 may repeat for each, or a portion, of scan positions withinthe measured magnetic field data received from sensors 110, 310, 410.Method 1800 may be implemented in a parallel or hierarchical manner,using multiple windows without departing from the scope hereof.

FIG. 19 shows a pairwise statistical comparison plot 1900 forcharacterizing ferromagnetic material. Pairwise statistical comparisonplot 1900 may be utilized by methods 1400, 1700, and 1800 tospecifically characterize the type of phenomenon, and in someembodiments the type of defect. That is, in addition to determining thata phenomenon occurs within the measured magnetic field data, methods1400, 1700, and 1800 may utilize plot 1900, or the data therefrom, todetermine what the phenomenon is (i.e. type of weld, type of defect,type of anomaly, etc.). Plot 1900 can be stored in server 160 or dataprocessing module 150 and can identify a library of phenomenon that canbeen seen in the field by systems 100, 200, 400, as well as howdifferent one known phenomenon is to another known phenomenon.

Pairwise statistical comparison plot 1900 is built by comparing themeasure of divergence for each pair of phenomena. Specifically, FIG. 19shows pairwise statistical comparison plot 1900 of features extractedfrom the measured magnetic field data 1920 (such as the angle betweenthe vector θ for different phenomena) versus modeled magnetic field 1930for ten different phenomena 1901-1910. In another embodiment, a finiteelement based model is used in place of modeled magnetic field 1930. Theten phenomena include for example three welds 1901, 1902, 1903, whichare examples of weld 535, FIG. 5; phenomenon 1904 which is a smalldefect; phenomenon 1905 which is a detectable defect, such as defect450, FIG. 4; and, phenomena 1906-1910 which include other miscellaneousanomalies. Each value in the matrix represents a numerical divergencebetween pairwise comparisons of measured and modeled magnetic field datafor each of the ten phenomena 1901-1910. For example, column 4 “1904”,row 1 “1901” represents a pairwise comparison of small defect 1904 toweld 1901. A difference between measured and modeled data is shown withlegend 1940. The entries in FIG. 19 are a measure of the statisticaldivergence between two phenomena, such as a weld and a defect. As such,in FIG. 19, phenomena 1901 and 1908 are separated by a small divergenceand are therefore relatively similar, when contrasted to phenomena 1901and 1905. Alternatively, phenomenon 1901 is more similar to phenomenon1908, than it is to phenomenon 1905.

The measure of divergence may be based on many variables, and more thanone variable may be used to build the pairwise statistical plot of FIG.19. For example, for two phenomena, we have two estimates of the vectorθ, or

,

. The angle between these vectors may be a measure of divergence. Thelarger the angles, the more distinct are the phenomena (e.g. the largerthe divergence), and vice versa. If that angle is not above a threshold,then the phenomena pair is not distinguishable. The threshold may bebased on the quality of the measurement, or the sensor noise variance orsignal to noise ratio. Other divergences may also be utilized, forexample, when non-parametric noise methods are preferred, divergencebetween histograms or rank histograms may be used. One example is theKullback Leibler divergence. Divergences derived from machine learningmethods are also possible.

To specifically characterize a detected phenomenon using plot 1900, dataprocessing module 150 (or server 160), implementing methods 1400, 1700,or 1800 may utilize a statistic from the test of equations (6) or (7),for instance. Take the case where the matrix F is zero (which could alsomean that the matrices S and F are aggregated). The likelihood ratio iscompared to a threshold, determining that a phenomenon of interest ispresent, as discussed above. In turn, data processing module 150 mayobtain the estimate of vector {circumflex over (θ)}, and compare it tothe value vector

, where e can be any of the events 1901 thru 1910. The comparison isbased on the angle between vector {circumflex over (θ)} and the givenvector

. The comparison yielding the smallest angle indicates the observedphenomenon.

Pairwise statistical plot 1900 may include a machine learning featurewhere, if the smallest angle between θ, and θ_e, for all events e isabove a certain threshold, then the answer would be “event or phenomenonnot seen before”.

It should be appreciated that the plot 1900 may be just one of manyplots analyzed by data processing module 150 (or server 160). Forexample, there may be multiple plots for each given window size. In sucha case, data processing module 150 may obtain multiple divergences forthe same pair and fuse at the higher decision level using decisionfusion methods, which may be learned using machine learning. Moreover,the system could fuse at the divergence level, and obtain a single fuseddiversion method, prior to decision.

FIGS. 20A, 20B, and 20C show three diagrams of exemplary schemes forcombining magnetic field data with data from other sensing modalities,such as ground penetrating radar, multimodal cameras, tomographicmeasurements, ultrasonic measurements, and active modulated magneticsignals for signal-to-noise ratio enhancement. FIGS. 20A-C are forexample diagrams of schemes implementing step 1442 of FIG. 14. Anydetails extracted from different measurements may be fused, at differentlevels, such as a measurement level, a data extraction level, or adetermination of defect versus non-defect level. FIG. 20A shows adiagram for fusing data from first, second, third and fourth modalities2010, 2020, 2030, 2040 at a measurement level in step 2050, followed byextracting phenomenon data in step 2060, determining defect versusnon-defect in step 2070, and optionally characterizing a defect and itslocation in step 2080.

FIG. 20B shows a diagram for fusing data at a phenomenon level.Specifically, phenomenon data are extracted for each of the fourmodalities in steps 2011, 2021, 2031, 2041 and fused in step 2061,followed by determining defect versus non-defect in step 2071 andoptionally characterizing a defect and its location in step 2080.

FIG. 20C shows a diagram for fusing data at a defect determining level.Specifically, a defect versus non-defect is determined in steps 2012,2022, 2032, 2042 from the four modalities 2010, 2020, 2030, 2040 and thedeterminations are fused in step 2072 to determine defect versusnon-defect, and optionally characterizing a defect and its location instep 2080.

Changes may be made in the above methods and systems without departingfrom the scope hereof. It should thus be noted that the matter containedin the above description or shown in the accompanying drawings should beinterpreted as illustrative and not in a limiting sense. The followingclaims are intended to cover all generic and specific features describedherein, as well as all statements of the scope of the present method andsystem, which might be said to fall there between.

What is claimed is:
 1. A method for characterizing a ferromagneticmaterial, comprising: providing a plurality of magnetometers arranged inan at least one-dimensional array, the array being positionable at astandoff distance along a z axis from the ferromagnetic material andtranslatable along an x axis, perpendicular to the z axis, to aplurality of scan positions along the ferromagnetic material, wherein atleast three magnetometers of the plurality of magnetometers are mutuallyspaced apart and disposed within the array along a first axis,perpendicular to the x and z axes; storing, in a memory, datarepresenting a deterministic physics-based model for missing metal; foreach scan position of the plurality of scan positions, calculating, by aprocessor, respective differences between a plurality of respectiveselect pairs of the magnetometers, wherein at least one select pair ofthe plurality of respective select pairs of the magnetometers comprisestwo non-adjacent magnetometers of the at least three magnetometersdisposed along the first axis, and each select pair of at least twoother select pairs of the plurality of respective select pairs of themagnetometers comprises two adjacent magnetometers of the at least threemagnetometers disposed along the first axis, and wherein each differenceof the respective differences is calculated by dividing by a distanceseparating the respective select pair of magnetometers; and comparing,by the processor, respective differences between the plurality ofrespective select pairs of the magnetometers with the data representingthe deterministic physics-based model for missing metal to identifymetal missing from the ferromagnetic material.